An in-depth theory: Jax and Kai'Sa are connected.
Whether their influence in each other's lives is small or large, I've put a lot of thought into this.
First, Jax and Kai'sa are both from the Shuriman continent and their lives have both have been directly effected by the void. They're exceptional fighters that have come face to face with the void and seemingly seek to destroy what remains of it on Runeterra.
Jax and Kai'Sa are BOTH considered monsters by the modern people of Runeterra. Kai'Sa because of her symbiosis, and Jax because of his unknown (From what we know, he could have been physically deformed by the void, but from this post it is heavily implied he wasn't corrupted. Other than this, in "Where Icathia Once Stood" Jax is described as "a shaven-headed giant with skin pockmarked by the ravages of a childhood illness" which could provide a possible alternative as to why he is considered a monster, especially since it is also heavily implied Jax is not of a known current race. So obviously modern humans would be a bit scared of a "giant" like Jax). But anyway, why is this relevant? Well, they're both monster hunters. So if they're hunting a void creature, who show up mainly in Shurima, they MOST LIKELY have caught wind of townspeople gossip of a "monster" lurking in certain areas, and they could have easily tracked each-other down with intentions of hunting a void creature, only to find that it was just someone who understood what it was like to be called a monster, when all you're trying to do is fight for your homeland.
Jax's current lore is based off of him searching for warriors, Kai'Sa would fit the bill PERFECTLY.
Now, I'm not saying they're actively integrated into each others lore's, but I'm almost certain they have at least met, it only makes sense.
In-game interactions are quite fickle, and I acknowledge that they aren't the most reliable source of information, but it just so happens that Kai'Sa has only 1 interaction with a champion who isn't related to her, void-born or the roles of adc/support. Yep, you guessed it: Jax. "A lamp's not going to do it Jax" "I guess I'm next Jax"
I'm gonna get a bit stretchy here, but the tone in which Kai'Sa speaks to Jax genuinely feels familiar and playful, and the fact that it JUST so happens that Jax is the only champion not currently ;) canonically connected to her in her special interactions. Putting everything together I genuinely believe they've met: A rioter also stated this: " There is a possibility that their paths could have crossed. She comments on what she sees/hears in game, but there's always the possibility of something more... "
The comment convinces me 100% that they have AT VERY LEAST crossed paths.
ALL ABOARD The #Kai'Jax ship is sailiiiinnnnggg